Question

Equal charges $q$ are placed at the four corners $A,B,C,D$ of a square of length $a.$ The magnitude of the force on the charge at $B$ will be:

• A. $\frac{3 q^{2}}{4 \pi \epsilon _{0} a^{2}}$
• B. $\frac{4 q^{2}}{4 \pi \epsilon _{0} a^{2}}$
• C. $\left(\frac{1 + 2 \sqrt{2}}{2}\right)\frac{q^{2}}{4 \pi \left(\epsilon \right)_{0} a^{2}}$
• D. $\left(2 + \frac{1}{\sqrt{2}}\right)\frac{q^{2}}{4 \pi \left(\epsilon \right)_{0} a^{2}}$

Answer 1

Answer: (C)

Total force on charge at $B$ is
$\overset{ \rightarrow }{F}_{B}=\overset{ \rightarrow }{F}_{A}+\overset{ \rightarrow }{F}_{C}+\overset{ \rightarrow }{F}_{D}$
$\left|\overset{ \rightarrow }{F}_{A}\right|=\left|\overset{ \rightarrow }{F}_{C}\right|=\frac{1}{4 \pi \epsilon _{0}}\frac{q^{2}}{a^{2}}$
$\left|\left(\overset{ \rightarrow }{F}\right)_{D}\right|=\frac{1}{4 \pi \left(\epsilon \right)_{0}}\frac{q^{2}}{\left(\right. a \sqrt{2} \left(\left.\right)^{2}}$
$\left|\overset{ \rightarrow }{F}_{A} + \overset{ \rightarrow }{F}_{C}\right|=\frac{\sqrt{2} q^{2}}{4 \pi \epsilon _{0} a^{2}}$
$\overset{ \rightarrow }{F}_{A}+\overset{ \rightarrow }{F}_{C}$ is along $\overset{ \rightarrow }{F}_{D}$
so $F_{B}=\frac{\sqrt{2} q^{2}}{4 \pi \left(\epsilon \right)_{0} a^{2}}+\frac{q^{2}}{4 \pi \left(\epsilon \right)_{0} \left(2 a^{2}\right)}$